Energy Flow Lines Research Articles

Numerical evaluation of Sommerfeld-type integrals for reflection and transmission of dipole radiation

A radiating electric dipole is located near the interface with a layer of material. The electric and magnetic fields reflect off the interface and transmit through the material. The exact solution of Maxwell’s equations can be found in terms of Sommerfeld-type integrals. These integrals have in general a singularity on the integration axis, and the integrands are extremely complicated functions of the parameters in the problem. We present a method for the computation of these integrals, and the corresponding electric and magnetic fields. Key to the solution is the splitting of the incident field in its traveling and evanescent contributions. With a change of variables, the singularities can be transformed away, and the method also greatly improves the accuracy and efficiency of the integration. We illustrate the feasibility of our approach with the computation of the flow lines of electromagnetic energy in the system. For such flow diagrams, a large number of integrals needs to be computed with reasonable accuracy. We show that in our approach even the smallest details in flow diagrams can be revealed. Program summaryProgram titles: CPiP-Auxiliary-1, CPiP-Auxiliary-2, CPiP-Field lines-1, CPiP-Field lines-2.CPC Library link to program files:http://dx.doi.org/10.17632/476n5ffkvv.1Licensing provisions: GPLv3.Programming language: Mathematica.Nature of problem: In near-field optics and nano-photonics, exact solutions of Maxwell’s equations are needed. Of particular interest are the reflected and transmitted electric and magnetic fields of dipole radiation by a layer of material. These solutions involve a large number of integrals, which need to be computed numerically. In the literature, these integrals are known as Sommerfeld-type integrals.Solution method: We split the integration range in two parts. The first part corresponds to traveling dipole waves and the second part results from evanescent dipole waves. In each region we make a (different) change of variables. The result of this transformation is that it removes a possible singularity on the integration axis, and it also has a tendency to smoothen out the integrand. There are 38 different types of integrals. Our method applies to all of them, and is self-contained. There is no need for tweaking of the programs for each of these, and no adjustments need to be made for different values of the parameters. The method is developed for the near field. For large distances to the source (the far field), asymptotic methods are available, and there would be no need for numerical integration.

Dipole radiation near a reflecting corner

We consider a radiating electric dipole, located near the joint of two orthogonal mirrors. The field lines of energy flow in the neighborhood of the dipole have an intriguing structure due to interference between the dipole radiation and the reflected light by the mirrors. Numerous singularities and vortices appear in the sub-wavelength region between the dipole and the surfaces. We present a method to find the locations of the vortices and singularities without regard to the details of the flow pattern. The radiation field induces a surface current density in the mirrors. The direction of the current is predominantly in the radial direction for a linear dipole, but it alternates between outgoing and incoming across singular curves. We show that the field line pattern expands with a phase velocity larger than the speed of light. For a circular dipole, there appears a spiral which runs inward. The current initially flows in along this spiral. Then the current leaves again along an outgoing spiral, which spirals inside the incoming spiral. Current can flow from one mirror to the other, and we show that the current always crosses the intersection line at a 90° angle.

Light transport and vortex-supported wave-guiding in micro-structured optical fibres.

In hydrodynamics, vortex generation upon the transition from smooth laminar flows to turbulence is generally accompanied by increased dissipation. However, vortices in the plane can provide transport barriers and decrease losses, as it happens in numerous geophysical, astrophysical flows and in tokamaks. Photon interactions with matter can affect light transport in ways resembling fluid dynamics. Here, we demonstrate significant impact of light vortex formation in micro-structured optical fibres on the energy dissipation. We show possibility of vortex formation in both solid core and hollow core fibres on the zero energy flow lines in the cladding. Through intensive numerical modelling using different independent approaches, we discovered a correlation between appearance of vortices and reduction of light leakage by three orders of magnitude, effectively improving wave guiding. This new effect potentially might have strong impact on numerous practical applications of micro-structured fibres. For instance, a strong light localization based on the same principle can also be achieved in the negative curvature hollow core fibres.

Energy flow of electric dipole radiation in between parallel mirrors

We have studied the energy flow patterns of the radiation emitted by an electric dipole located in between parallel mirrors. It appears that the field lines of the Poynting vector (the flow lines of energy) can have very intricate structures, including many singularities and vortices. The flow line patterns depend on the distance between the mirrors, the distance of the dipole to one of the mirrors and the angle of oscillation of the dipole moment with respect to the normal of the mirror surfaces. Already for the simplest case of a dipole moment oscillating perpendicular to the mirrors, singularities appear at regular intervals along the direction of propagation (parallel to the mirrors). For a parallel dipole, vortices appear in the neighbourhood of the dipole. For a dipole oscillating under a finite angle with the surface normal, the radiating tends to swirl around the dipole before travelling off parallel to the mirrors. For relatively large mirror separations, vortices appear in the pattern. When the dipole is off-centred with respect to the midway point between the mirrors, the flow line structure becomes even more complicated, with numerous vortices in the pattern, and tiny loops near the dipole. We have also investigated the locations of the vortices and singularities, and these can be found without any specific knowledge about the flow lines. This provides an independent means of studying the propagation of dipole radiation between mirrors.

Energy flows in rock mass under tidal deformation

Under analysis is the stress state of an elliptical domain under varying loading conditions. The energy flow lines are plotted. The paper demonstrates the effect of the boundary conditions on the shape of the flow lines.

Propagation of magnetic dipole radiation through a medium.

An oscillating magnetic dipole moment emits radiation. We assume that the dipole is embedded in a medium with relative permittivity ϵr and relative permeability μr, and we have studied the effects of the surrounding material on the flow lines of the emitted energy. For a linear dipole moment in free space the flow lines of energy are straight lines, coming out of the dipole. When located in a medium, these field lines curve toward the dipole axis, due to the imaginary part of μr. Some field lines end on the dipole axis, giving a nonradiating contribution to the energy flow. For a rotating dipole moment in free space, each field line of energy flow lies on a cone around the axis perpendicular to the plane of rotation of the dipole moment. The field line pattern is an optical vortex. When embedded in a material, the cone shape of the vortex becomes a funnel shape, and the windings are much less dense than for the pattern in free space. This is again due to the imaginary part of μr. When the real part of μr is negative, the field lines of the vortex swirl around the dipole axis opposite to the rotation direction of the dipole moment. For a near-single-negative medium, the spatial extent of the vortex becomes huge. We compare the results for the magnetic dipole to the case of an embedded electric dipole.

The giant dipole vortex

The field lines of energy flow of radiation emitted by an oscillating electric dipole in free space are either straight lines (linear dipole) or they form a vortex (rotating dipole). When the dipole is embedded in a material, the properties of the medium affect the direction of energy flow. Damping due to the imaginary part of the relative permittivity makes the field lines curve for the case of a linear dipole, and for a rotating dipole, the shape of the vortex is altered. In addition, a negative value of the real part of has the effect that the rotation direction of the vortex reverses for the case of a rotating dipole. The value of the relative permeability has in general not much effect on the redistribution of the direction of energy propagation. We show that a dramatic effect occurs when the embedding material is near-single-negative (both and approximately real, and the real parts of opposite sign). The curving of field lines is in general a sub-wavelength phenomenon. For near-single-negative materials, however, this curving extends over large distances from the dipole. In particular, the small free-space vortex of a rotating dipole becomes a vortex of enormous dimensions when the radiation is emitted into a near-single-negative material.

Description of Classical and Quantum Interference in View of the Concept of Flow Line

Bohmian mechanics, a hydrodynamic formulation of quantum mechanics, relies on the concept of trajectory, which evolves in time in compliance with dynamical information conveyed by the wave function. Here this appealing idea is considered to analyze both classical and quantum interference, thus providing an alternative and more intuitive framework to understand the time-evolution of waves, either in terms of the flow of energy (for mechanical waves, sound waves, electromagnetic waves, for instance) or, analogously, the flow of probability (quantum waves), respectively. Furthermore, this procedure also supplies a more robust explanation of interference phenomena, which currently is only based on the superposition principle. That is, while this principle only describes how different waves combine and what effects these combinations may lead to, flow lines provide a more precise explanation on how the energy or probability propagate in space before, during and after the combination of such waves, without dealing with them separately (i.e., the combination or superposition is taken as a whole). In this sense, concepts such as constructive and destructive interference, typically associated with the superposition principle, physically correspond to more or less dense swarms of (energy or probability) flow lines, respectively. A direct consequence of this description is that, when considering the distribution of electromagnetic energy flow lines behind two slits, each one covered by a differently oriented polarizer, it is naturally found that external observers' information on the slit crossed by single photons (understood as energy parcels) is totally irrelevant for the existence of interference fringes, in striking contrast with what is commonly stated and taught.

Energy flows in problems on influence of pressing tool on half-plane

The authors have plotted lines of energy flow in the problems on uniform vertical force applied at the boundary of an elastic half-plane and on indentation of a smooth pressing tool in a rigid-plastic medium. The influence of stresses on displacement velocities is analyzed. As an example, formation of compaction zone under ram block is studied.

Energy transport in the near field of an electric dipole near a layer of material

We consider an oscillating electric dipole, embedded in a uniform medium with relative permittivity and relative permeability . The dipole is located near an interface with a layer with uniform material parameters and , and the second interface borders a uniform medium with parameters and . We have obtained the solutions for the electric and magnetic fields in the various regions, without any restrictions on the parameters and for any state of oscillation of the dipole (elliptical, in general). The solution involves a set of auxiliary functions, which are given as integral representations containing the Fresnel coefficients for plane waves. With this solution, the field lines of energy flow can be obtained, and we have considered the flow pattern for the simple case of a dipole oscillating perpendicular to the interface. When the material of the layer is optically thicker than the embedding medium of the dipole, energy flows more or less along straight lines. At an interface, the field lines refract, similar to optical rays. When the layer material is optically thinner, the energy flow lines curve. A portion of the energy that propagates toward the interface bends away from it before reaching the interface. Other field lines of energy flow cross the interface, but then return to the area of the dipole by crossing the interface again. This leads to an oscillation of energy back and forth through the interface. In the neighborhood of this oscillation, a concentric set of vortex tori appears.

Photon field and energy flow lines behind a circular disc

In order to evaluate electromagnetic energy flow (EME) lines behind a circular disc illuminated by monochromatic laser light, the Rayleigh–Sommerfeld integral is transformed to a simpler form. EME flow lines, interpreted as photon trajectories, explain the change of the intensity pattern with the distance from a circular disc very well, including the Poisson–Arago spot in the center. Possible applications of these results on the study of Poisson–Arago spot experiments with matter beams, are indicated and explained.

Weak measurement and its experimental realisation

The relationship between the real part of the weak value of the momentum operator at a post selected position is discussed and the meaning of the experimentally determined stream-lines in the Toronto experiment of Kocsis et al is re-examined. We argue against interpreting the energy flow lines as photon trajectories. The possibility of performing an analogous experiment using atoms is proposed in order that a direct comparison can be made with the trajectories calculated by Philippidis, Dewdney and Hiley using the Bohm approach.

Vortex strings in electric dipole radiation near a mirror

The energy flow pattern of the radiation emitted by an oscillating electric dipole near a mirror has a complicated structure, including numerous singularities and vortices. We consider the flow lines of energy in the plane through the surface normal and the oscillation direction of the dipole. It is shown that the vortices are due to the vanishing of the magnetic field at their centers. The locations of the vortices have the appearance of beads on strings, and there are four such strings. The rotation direction of the energy flow for each vortex on a given string is the same. There are two strings with clockwise rotation and two strings with counterclockwise rotation. Field lines of energy flow either start or end at the center of a vortex. For a given string, field lines end at each vortex or field lines start at each vortex. There are two strings on which field lines end at the centers of the vortices, and there are two strings on which field lines start inside the vortices.

A Diffraction Model for Acoustical Ray-Tracing Based on the Energy Flow Lines Concept

A Diffraction Model for Acoustical Ray-Tracing Based on the Energy Flow Lines Concept

Energy sources of rockbursts

The authors analyze stress-strain state of elastic plane with circular void in varied loading conditions, draw energy flow lines, highlight effect of clustered seismic events on the flow line shapes, and propose definition of rockburst probability.

Propagation of Electric Dipole Radiation through a Medium

When electromagnetic energy propagates through a material medium, the paths of energy flow may be altered, as compared to propagation in free space. We consider radiation emitted by an electric dipole, embedded in a medium with permittivity and permeability . For a linear dipole in free space, the field lines of energy flow are straight, but when the imaginary part of is finite, the field lines in the material become curves in the near field of the dipole. Therefore, the energy flow is redistributed due to the damping in the material. For a circular dipole in free space, the field lines of energy flow wind around the axis perpendicular to the plane of rotation of the dipole moment. When has an imaginary part, this flow pattern is altered drastically. Furthermore, when the real part of is negative, the direction of rotation of the flow lines reverses. In that case, the energy in the field rotates opposite to the direction of rotation of the dipole moment. It is indicated that in metamaterials with a negative index of refraction this may lead to an observable effect in the far field.

Optical Scully vortex and its spatial evolution

The structure of an optical vortex formed in a partially coherent Laguerre-Gauss laser beam was considered. The main object of study was the recorded vector field of wavefront tilts that consisted of the vortical and potential components. It was found that the vortical motion weakened as the coherence decreased. Main regularities in the behavior of the vortical component can be described by the Scully vortex model of vortical liquid flow. In the spatial evolution, the potential component of tilts may alternate the sign, thus determining the direction of energy flow to the center or to the periphery of the vortex. Energy flow lines in the beam demonstrate the pattern of decay of an optical vortex similar to the pattern of decaying vortical motion in viscous liquid.

Stream function for optical energy flow

A formula is presented for the stream functionS corresponding to optical fields in the plane that can be represented as a superposition ofplane waves with the same transverse wavenumbers, including evanescent waves. Energyflow lines (parallel to the Poynting vector) are easily computed as contour lines ofS. Optical vortices (including higher-order vortices) correspond to extrema ofS.

Damping of the dipole vortex

When a circular electric dipole moment, rotating in the x-y plane, is embedded in a material with relative permittivity ε(r) and relative permeability μ(r), the field lines of energy flow of the emitted radiation are dramatically influenced by the surrounding material. For emission in free space, the field lines swirl around the z axis and lie on a cone. The direction of rotation of the field lines around the z axis is the same as the direction of rotation of the dipole moment. We found that when the real part of ε(r) is negative, the rotation of the field lines changes direction, and hence the energy counter-rotates the dipole moment. When there is damping in the material, due to an imaginary part of ε(r), the cone turns into a funnel, and the density of the field lines diminishes near the location of the source. In addition, all radiation is emitted along the z axis and the x-y plane, whereas for emission in free space, the radiation is emitted in all directions. It is also shown that the displacement of the dipole image in the far field depends on the material parameters and that the shift can be much larger than the shift of the image in free space.

Redistribution of energy flow in a material due to damping

The field lines of energy flow of the radiation emitted by a linear dipole in free space are straight lines, running radially outward from the source. When the dipole is embedded in a medium, the field lines are curves when the imaginary part of the relative permittivity is finite. It is shown that due to the damping in the material all radiation is emitted in directions perpendicular to the dipole axis, whereas for a dipole in free space the radiation is emitted in all directions except along the dipole axis. It is also shown that some field lines in the near field form semiloops. Energy flowing along these semiloops is absorbed by the material and does not contribute to the radiative power in the far field.